principled models in the analysis of social relationships

the case of agonistic dominance

Christof Neumann

Cognitive Ethology Lab, German Primate Center

defining ad hockery

the practice of reacting to what happens or is needed at a particular time, rather than planning in a way that suits all possible situations

  • lack of formal grounding
  • coming up with something just like that

    • etwas aus dem Ärmel schütteln

    • faire qc. en un simple claquement de doigts

ad hockery is everywhere common

  • we should ground things in substantial theory

  • or at least try to get more precise definitions and operationalizations

  • what is steepness, exactly?

  • what is degree of female dominance, exactly?

  • what is bond strength, exactly?

what can we do?

  • in a nutshell we need to parameterize things = put our analyses on a solid basis

  • parameter: unknown quantity that governs the behavior of a probability distribution or a causal mechanism

  • focus today: parameterize features of sociality (dominance)

  • ideally, these features are tractable (no AI here)

  • with parameters:

    • we can criticize our models, improve them, use them as pieces in joint models

    • we can reverse-engineer -> simulate the data generating process

analysis of dominance interactions

  • iterative algorithms over the entire interaction network (e.g. I&SI, de Vries (1998), ADAGIO, Douglas, Ngonga Ngomo, and Hohmann (2017))

  • based on dyadic proportions (e.g. DS, David (1987), CBI, Clutton-Brock et al. (1979))

  • sequential analysis (e.g. Elo-rating, Elo (1978), Glicko, Glickman (1999))

  • ‘exotics’ (e.g. Fushing, McAssey, and McCowan (2011), Adams (2005))

workflow

  • interaction data (either matrix or sequence): a subset of true interactions
  • individual-level
    • dominance hierarchy with ordinal ranks
    • and/or dominance scores/ratings
  • group-level
    • linearity, transitivity
    • steepness
    • degree of female dominance

interaction data you say?

Date Time winner loser Draw
2000-01-01 12:56 x r FALSE
2000-01-02 17:03 v o FALSE
2000-01-03 11:38 r o FALSE
2000-01-04 13:00 r v FALSE
2000-01-05 12:33 q v FALSE
2000-01-06 16:22 o v FALSE
2000-01-07 06:42 o v FALSE
2000-01-08 09:13 o q FALSE
2000-01-09 13:17 q x FALSE
2000-01-10 14:58 q r FALSE
2000-01-11 14:50 o q FALSE
2000-01-12 12:32 r x FALSE
2000-01-13 09:20 r x FALSE
2000-01-14 12:20 o r FALSE
2000-01-15 17:41 v r FALSE
2000-01-16 11:53 o q FALSE
2000-01-17 17:45 q v FALSE
2000-01-18 17:09 q r FALSE
2000-01-19 06:06 v o FALSE
2000-01-20 10:08 o v FALSE
2000-01-21 11:14 r x FALSE
2000-01-22 06:20 o x FALSE
2000-01-23 14:36 v q FALSE
2000-01-24 12:40 q v FALSE
2000-01-25 12:24 v x FALSE
2000-01-26 08:45 q o FALSE
2000-01-27 11:36 r v FALSE
2000-01-28 07:36 q o FALSE
2000-01-29 15:33 o v FALSE
2000-01-30 16:41 o x FALSE
2000-01-31 15:24 r v FALSE
2000-02-01 12:43 v x FALSE
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-05 15:34 o x FALSE
2000-02-06 06:05 v q FALSE
2000-02-07 16:09 q o FALSE
2000-02-08 16:41 o x FALSE
2000-02-09 15:37 q v FALSE
2000-02-10 15:46 q x FALSE
2000-02-11 13:19 q v FALSE
2000-02-12 15:27 x o FALSE
2000-02-13 10:19 v x FALSE
2000-02-14 17:57 q o FALSE
2000-02-15 12:43 o v FALSE
2000-02-16 13:56 o r FALSE
2000-02-17 10:22 o r FALSE
2000-02-18 17:05 o q FALSE
2000-02-19 11:56 r v FALSE
2000-02-20 13:39 r x FALSE
2000-02-21 06:43 o x FALSE
2000-02-22 08:24 o r FALSE
2000-02-23 15:54 o r FALSE
2000-02-24 08:55 x v FALSE
2000-02-25 16:05 r v FALSE
2000-02-26 06:57 v x FALSE
2000-02-27 11:38 o v FALSE
2000-02-28 11:50 q v FALSE
2000-02-29 17:41 o v FALSE
2000-03-01 09:05 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-03 10:31 q x FALSE
2000-03-04 06:13 o x FALSE
2000-03-05 08:01 v x FALSE
2000-03-06 11:44 o x FALSE
2000-03-07 08:17 o v FALSE
2000-03-08 12:21 v x FALSE
2000-03-09 08:13 o r FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE
2000-03-12 06:50 q r FALSE
2000-03-13 14:50 r x FALSE
2000-03-14 17:39 q v FALSE
2000-03-15 07:05 v o FALSE
Date Time winner loser Draw
2000-01-01 12:56 x r FALSE
2000-01-02 17:03 v o FALSE
2000-01-03 11:38 r o FALSE
2000-01-04 13:00 r v FALSE
2000-01-05 12:33 q v FALSE
2000-01-06 16:22 o v FALSE
2000-01-07 06:42 o v FALSE
2000-01-08 09:13 o q FALSE
2000-01-09 13:17 q x FALSE
2000-01-10 14:58 q r FALSE
2000-01-11 14:50 o q FALSE
2000-01-12 12:32 r x FALSE
2000-01-13 09:20 r x FALSE
2000-01-14 12:20 o r FALSE
2000-01-15 17:41 v r FALSE
2000-01-16 11:53 o q FALSE
2000-01-17 17:45 q v FALSE
2000-01-18 17:09 q r FALSE
2000-01-19 06:06 v o FALSE
2000-01-20 10:08 o v FALSE
2000-01-21 11:14 r x FALSE
2000-01-22 06:20 o x FALSE
2000-01-23 14:36 v q FALSE
2000-01-24 12:40 q v FALSE
2000-01-25 12:24 v x FALSE
2000-01-26 08:45 q o FALSE
2000-01-27 11:36 r v FALSE
2000-01-28 07:36 q o FALSE
2000-01-29 15:33 o v FALSE
2000-01-30 16:41 o x FALSE
2000-01-31 15:24 r v FALSE
2000-02-01 12:43 v x FALSE
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-05 15:34 o x FALSE
2000-02-06 06:05 v q FALSE
2000-02-07 16:09 q o FALSE
2000-02-08 16:41 o x FALSE
2000-02-09 15:37 q v FALSE
2000-02-10 15:46 q x FALSE
2000-02-11 13:19 q v FALSE
2000-02-12 15:27 x o FALSE
2000-02-13 10:19 v x FALSE
2000-02-14 17:57 q o FALSE
2000-02-15 12:43 o v FALSE
2000-02-16 13:56 o r FALSE
2000-02-17 10:22 o r FALSE
2000-02-18 17:05 o q FALSE
2000-02-19 11:56 r v FALSE
2000-02-20 13:39 r x FALSE
2000-02-21 06:43 o x FALSE
2000-02-22 08:24 o r FALSE
2000-02-23 15:54 o r FALSE
2000-02-24 08:55 x v FALSE
2000-02-25 16:05 r v FALSE
2000-02-26 06:57 v x FALSE
2000-02-27 11:38 o v FALSE
2000-02-28 11:50 q v FALSE
2000-02-29 17:41 o v FALSE
2000-03-01 09:05 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-03 10:31 q x FALSE
2000-03-04 06:13 o x FALSE
2000-03-05 08:01 v x FALSE
2000-03-06 11:44 o x FALSE
2000-03-07 08:17 o v FALSE
2000-03-08 12:21 v x FALSE
2000-03-09 08:13 o r FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE
2000-03-12 06:50 q r FALSE
2000-03-13 14:50 r x FALSE
2000-03-14 17:39 q v FALSE
2000-03-15 07:05 v o FALSE
Date Time winner loser Draw
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-08 16:41 o x FALSE
2000-02-18 17:05 o q FALSE
2000-02-21 06:43 o x FALSE
2000-02-26 06:57 v x FALSE
2000-02-29 17:41 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-08 12:21 v x FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE

matrix shenanigans

  • David’s scores are based on dyadic winning proportions

 

ID DS normDS
o 2.12 2.42
v 0.45 2.09
q 0.44 2.09
r 0.00 2.00
x -3.02 1.40

Elo-rating

  • sequential processing: winner’s rating increases, loser’s rating decreases
  • deterministic algorithm that requires start ratings and \(k\)
  • magnitude of change depends on \(k\) and the probability of the winner to win
  • classic versions: start ratings and \(k\) are set to fixed values

winning probability you say?

\[ \begin{align} p_{\text{win}} &= \frac{1}{1 - (1 + 10 ^ {(\text{Elo}_A - \text{Elo}_B)/400})}\\ p_{\text{win}} &= \frac{1}{1 + \text{exp}(-0.01(\text{Elo}_B - \text{Elo}_A))}\\ p_{\text{win}} &= \text{normal CDF}(\frac{(\text{Elo}_B - \text{Elo}_A)}{200 \times \sqrt{2}}) \end{align} \]

Elo (1978); Albers and de Vries (2001); Neumann et al. (2011); Franz et al. (2015); Goffe, Fischer, and Sennhenn-Reulen (2018); Sánchez-Tójar, Schroeder, and Farine (2018);

rhetoric questions

  • how would you simulate:
    • a dominance network with steepness 0.86?
    • grooming network with 12% dyads having larger sociality index values than 3.2?
  • if we don’t have a formal underlying model, simulation is impossible (or in itself ad hockery and/or relies on brute force)
  • coming up: one more slide of complaining

more examples of ad hoc solutions in the context of dominance analyses

  • stop because we think we have not observed enough interactions
  • missing dyads? individuals without any observed interactions?
  • what about tied interactions? what about coalitions?
  • observation bias?
  • steepness = regression of standardized scores by ordinal rank
  • randomizing the interaction sequence, really?
  • summing winning probabilities, really?
  • just run a bunch of methods and average?

Sánchez-Tójar, Schroeder, and Farine (2018); Hart et al. (2022)

progress: Elo-rating 2.0

  • set different \(k\) values depending on intensity, set differential start values for groups of animals
  • get ML estimates for start values and \(k\)
Newton-Fisher (2017); Franz et al. (2015)

Elo meets Bayes

  • Bayesian estimation of \(k\), start values and their SD
  • what predicts ratings (age in cueing systems, relatedness)
  • embrace (and use) the uncertainty in rating estimates
Goffe, Fischer, and Sennhenn-Reulen (2018); Neumann and Fischer (2023b)

solution: models with benefits

  • fit principled models, which reflect theory and data generating process

  • Bayesian framework, with all its benefits (and challenges and complexities)

principled models: dominance

principled models: dominance

principled models: dominance

principled models: dominance

principled models: dominance

ad hoc models: bond strength (and other features)

  • SI, CSI, DSI

  • Hinde Index

  • Reciprocity Index

  • SRI, HWI

Silk, Altmann, and Alberts (2006); Silk, Cheney, and Seyfarth (2013)

principled models: bond strength

Hart et al. (2023); Neumann and Fischer (2023a); Ross, McElreath, and Redhead (2024)

uncertainty in parameters

  • this is what proportions of 0 might actually look like (through the Bayesian lens)

  • precision can increase with increasing observation effort

\(Pr(\text{0}) \sim \text{Binomial}([1, 10, 100], \theta)\)

uncertainity in parameters

  • and proportions of 0.5, and it works for rates too

background

steepness measures the degree to which individuals differ from each other in winning dominance encounters

  • incidentally: nothing about ranking in this definition…

  • ‘classic steepness’ is coming from David’s scores

de Vries, Stevens, and Vervaecke (2006)

David’s scores

  • are based on proportions of dyadic winning/losing

David (1987); de Vries, Stevens, and Vervaecke (2006); Neumann and Fischer (2023b)

from DS to steepness

  • propagate uncertainty from DS -> steepness

David (1987); de Vries, Stevens, and Vervaecke (2006); Neumann and Fischer (2023b)

female dominance

the percentage of males dominated by each female (the degree of female dominance)

  • there is nothing about ranking in that definition/operationalization
  • being dominant or not manifests at the level of the dyad
Drews (1993); Kappeler et al. (2022)

a principled model of female dominance

\[ \begin{align} \text{interactions won}_i &\sim \operatorname{Binomial}(\operatorname{logit}^{-1}(\mu_{i}), \text{interactions}_i) \\ \mu_{i} &= b_0 + b_{\text{g}_i} + b_{\text{f}_i} + b_{\text{m}_i} \\ b_{\text{g}_i} &\sim \operatorname{Normal}(0, \sigma_g) \\ b_{\text{f}_i} &\sim \operatorname{Normal}(0, \sigma_f) \\ b_{\text{m}_i} &\sim \operatorname{Normal}(0, \sigma_m) \\ \sigma_g &\sim \operatorname{Exponential}(1) \\ \sigma_f &\sim \operatorname{Exponential}(1) \\ \sigma_m &\sim \operatorname{Exponential}(1) \end{align} \]

a principled model of female dominance

  • it’s a glorified (but principled) way of saying we calculated proportions
  • intercept-only model with varying intercepts for group, female and male
  • actual values (‘BLUPS’) for the varying intercepts for group is the FDI
  • all parameters have priors and posteriors

a principled model of female dominance

 

direct comparison

  • the extremes match fairly well

  • discrepancies in the mid-range

Kappeler et al. (2022)

posterior predictive checks (aka Bayesian glory)

  • wouldn’t it be cool if we could try to predict/simulate interaction outcomes?
  • and compare those predictions to what we actually observed?

model in model

  • how is degree of female dominance associated with interaction features?
  • use posterior of FDI as predictor (not its point estimate)

\[ \begin{align} \text{aggr. act}_i &\sim \operatorname{Bernoulli}(\operatorname{logit}^{-1}(\mu_{i})) \\ \mu_{i} &= \mathcal{A}_i + \mathcal{B}_i * \text{FDI}_i + ... \\ \mathcal{A}_i & = \alpha + \alpha_{\textrm{winner}[i]} + \alpha_{\textrm{loser}[i]} + \alpha_{\textrm{dyad}[i]} + \alpha_{\textrm{group}[i]} & \text{intercept}\\ \mathcal{B}_i & = \beta + \beta_{\textrm{winner}[i]} + \beta_{\textrm{loser}[i]} + \beta_{\textrm{dyad}[i]} + \alpha_{\textrm{group}[i]} & \text{slope for FDI}\\ \text{ints. won}_j &\sim \operatorname{Binomial}(\operatorname{logit}^{-1}(\nu_{j}), \text{ints.}_j) \\ \nu_{j} &= b_0 + \mathbf{\text{FDI}_j} + b_{\text{f}_j} + b_{\text{m}_j} \end{align} \]

female female

  • very similar for male-male and mixed dyads

compare slopes

  • difference in width of posteriors is notable

reproductive success in crested macaque females

  • the association between sociality and reproductive output
  • both grooming and Elo rating are incorporated alongside their measurement error

modeling reproductive success

\[ \begin{align} \text{birth} &\sim \text{Bernoulli}(\text{logit}^{-1}(\mu))\\ \mu &= b_0 + b_1 * \text{E} + b_2 * \text{G}\\ \text{dom. interactions} &\sim f(\text{start ratings}, k)\\ \text{E} &= f(\text{dom. interactions}, \text{start ratings}, k)\\ \text{groom} &\sim \text{Binomial}(\text{tot. point samples}, \text{logit}^{-1}(G))\\ G &\sim \text{Normal}(0, 1) \end{align} \]

  • sample: 456 female-years, 106 females
Duboscq et al. (2023)

uncertainty in social features

  • uncertainty in grooming proportion decreases with more observation effort
  • uncertainty in Elo-rating decreases with more observed interactions

consequences

  • slope in error model is wider (as expected)
  • slope in error model is not pulled toward zero (somewhat surprising)

consequences

  • no change in inference between error model and simpler model without error

wrap up

  • think about the data generating process
  • ask what can I do? what do I want to do?
  • formalize your model as much as possible before fitting it to data
  • uncertainty propagation and dealing with measurement errors is an important aspect of a principled workflow

wrap up

  • we work a lot with proxies of vague theoretical provenance
  • most things we measure, we measure with error
  • but we typically give both these aspects not too much thought

wrap up

  • the systems we study are complex
  • statistics is hard
  • that shouldn’t stop us from trying

thank you

https://xkcd.com/2456

references

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Albers, Paul C H, and Han de Vries. 2001. “Elo-Rating as a Tool in the Sequential Estimation of Dominance Strengths.” Animal Behaviour 61: 489–95. https://doi.org/10.1006/anbe.2000.1571.
Clutton-Brock, Timothy H, Steve D Albon, Robert M Gibson, and Fiona E Guinness. 1979. “The Logical Stag: Adaptive Aspects of Fighting in Red Deer (Cervus Elaphus L.).” Animal Behaviour 27: 211–25. https://doi.org/10.1016/0003-3472(79)90141-6.
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Douglas, Pamela Heidi, Axel-Cyrille Ngonga Ngomo, and Gottfried Hohmann. 2017. “A Novel Approach for Dominance Assessment in Gregarious Species: ADAGIO.” Animal Behaviour 123: 21–32. https://doi.org/10.1016/j.anbehav.2016.10.014.
Drews, Carlos. 1993. “The Concept and Definition of Dominance in Animal Behaviour.” Behaviour 125: 283–313. https://doi.org/10.1163/156853993X00290.
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Fushing, Hsieh, Michael P McAssey, and Brenda J McCowan. 2011. “Computing a Ranking Network with Confidence Bounds from a Graph-Based Beta Random Field.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467: 3590–3612. https://doi.org/10.1098/rspa.2011.0268.
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Goffe, Adeelia S, Julia Fischer, and Holger Sennhenn-Reulen. 2018. “Bayesian Inference and Simulation Approaches Improve the Assessment of Elo-ratings in the Analysis of Social Behaviour.” Methods in Ecology and Evolution 9: 2131–44. https://doi.org/10.1111/2041-210X.13072.
Hart, Jordan D A, Michael N Weiss, Lauren J N Brent, and Daniel W Franks. 2022. “Common Permutation Methods in Animal Social Network Analysis Do Not Control for Non-Independence.” Behavioral Ecology and Sociobiology 76: 151. https://doi.org/10.1007/s00265-022-03254-x.
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Kappeler, Peter M., Elise Huchard, Alice Baniel, Charlotte Canteloup, Marie J. E. Charpentier, Leveda Cheng, Eve Davidian, et al. 2022. “Sex and Dominance: How to Assess and Interpret Intersexual Dominance Relationships in Mammalian Societies.” Frontiers in Ecology and Evolution 10: 918773. https://doi.org/10.3389/fevo.2022.918773.
Neumann, Christof, Julie Duboscq, Constance Dubuc, Andri Ginting, Ade Maulana Irwan, Muhammad Agil, Anja Widdig, and Antje Engelhardt. 2011. “Assessing Dominance Hierarchies: Validation and Advantages of Progressive Evaluation with Elo-rating.” Animal Behaviour 82: 911–21. https://doi.org/10.1016/j.anbehav.2011.07.016.
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———. 2023b. “Extending Bayesian Elo-rating to Quantify the Steepness of Dominance Hierarchies.” Methods in Ecology and Evolution 14: 669–82. https://doi.org/10.1111/2041-210X.14021.
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Sánchez-Tójar, Alfredo, Julia Schroeder, and Damien R Farine. 2018. “A Practical Guide for Inferring Reliable Dominance Hierarchies and Estimating Their Uncertainty.” Journal of Animal Ecology 87: 594–608. https://doi.org/10.1111/1365-2656.12776.
Silk, Joan B, Jeanne Altmann, and Susan C Alberts. 2006. “Social Relationships Among Adult Female Baboons (Papio Cynocephalus) I. Variation in the Strength of Social Bonds.” Behavioral Ecology and Sociobiology 61: 183–95. https://doi.org/10.1007/s00265-006-0249-2.
Silk, Joan B, Dorothy L Cheney, and Robert M Seyfarth. 2013. “A Practical Guide to the Study of Social Relationships.” Evolutionary Anthropology 22: 213–25. https://doi.org/10.1002/evan.21367.